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N-dimensional sparse arrays


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N-dimensional sparse arrays



26 Dec 2010 (Updated )

Creates an N-dimensional sparse array object, for arbitrary N.

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This submission defines a class of N-dimensional sparse arrays for N possibly greater than 2. However, it should really be thought of as a way of starting with an ordinary MATLAB sparse matrix and reshaping it to have N dimensions. In other words, the sparse data must first be able to exist as an ordinary 2D MATLAB sparse matrix before being made N-dimensional. In fact, if the intended array has dimensions MxNxP...YxZ, then the class will store it internally as an ordinary 2D sparse array of dimensions (M*N*P*...*Y)xZ. This leads to certain memory strains when using large numbers of dimensions. I find the class useful mainly for moderate dimensional things like edge detection in 3D imaging, where you often want to hold a sparse 3D edge map.
  S=ndSparse(X) where X is an ordinary MATLAB sparse matrix converts X into an ndSparse object. S can be reshaped into an N-dimensional sparse array using its RESHAPE method, for arbitrary N.
  S=ndSparse(X,[M,N,P,...]) is equivalent to reshape(ndSparse(X),[M,N,P,...]).
The class also has a variety of static methods that can be used to construct instances of the class. For example,,Values,[m,n,p,...],nzmax)

lets you generate an N-dimensional sparse array from a table of explicit entries. This is a generalization to N dimensions of S=sparse(i,j,s,m,n,nzmax).

Other such methods include:


    >> [1 1 1; 2 1 1;2 2 2] , [50,60 70]) %Builds a 2x2x2 sparse array from table

     A(:,:,1) =

        (1,1) 50
        (2,1) 60

     A(:,:,2) =

        (2,2) 70

Many of the same manipulations common to ordinary multidimensional MATLAB full arrays can be performed on the sparse 3D array A generated above. It can be permuted, summed, concatentated, and so forth e.g.,

    >> B=sum( permute([A,A+10], [3,2,1]) ,2)

     B(:,:,1) =

       (1,1) 120
       (2,1) 20

     B(:,:,2) =

       (1,1) 140
       (2,1) 160

Other overloaded methods include BSXFUN, REPMAT, CIRCSHIFT, CONVN, FLIPDIMS, SQUEEZE, SHIFTDIM and many more. Type "methods ndSparse" for a full list and use "help ndSparse.methodname" to get details of usage.

When browsing the list of methods, note that certain common operations have different implementations, optimized for different situations. Specifically, SUM, ANY,ALL, MIN, MAX... have alternative implementations SUMML, ANYML, ALLML, MINML, MAXML which are optimized for "low-dimensional" ndSparse objects OBJ. Here, low-dimensional means that a normal N-column MATLAB sparse matrix won't consume too much memory on your platform for N=MAX(NUMEL(OBJ)./SIZE(OBJ)).

Another feature of the class is that bi-operand operations are allowed between ndSparse objects and MATLAB objects of any numeric type (single, uint16, etc...). This is not true of ordinary MATLAB sparse matrices, as of R2010b.
    >> C=eye(2,'single')*B(:,:,2)

     C =

       (1,1) 140
       (2,1) 160

  >> whos A B C

  Name Size Bytes Class Attributes

  A 2x2x2 136 ndSparse
  B 2x1x2 140 ndSparse
  C 2x1 104 ndSparse
To convert back to an ordinary n-D full array, use the class' overloaded FULL method. To convert to a normal 2D sparse matrix, use the methods SPARSE or SPARSE2D. For example, SPARSE2D will convert an MxNxPx...xQ ndSparse array to the two dimensional (M*N*P*...)xQ sparse matrix in native MATLAB form.


Sub2allind inspired this file.

MATLAB release MATLAB 7.11 (R2010b)
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Comments and Ratings (76)
21 Oct 2014 Tom  
07 Mar 2014 Caglayan Dicle  
01 May 2013 Matt J

Do you still have get that error? What version of MATLAB are you using? And what version of ndSparse?

You seem to be the only one with this problem, I'm afraid. It doesn't readily reproduce on my machine and I haven't received similar complaints.

01 May 2013 Lauren

Hi Matt-
Not sure why my previous comment no longer appears, but I had mentioned that I get an error message when attempting to run ndstest -- the error message being something about the inputs to kron being of the wrong type, coming from line 882 of ndtest.


30 Apr 2013 Matt J

Lauren: not clear what you meant by your "comment below". As far as I can tell, you posted no prior comments.

30 Apr 2013 Lauren

I realized that (for my comment below) the problem is that one of the inputs being sent to kron is ndSparse, which appears to not be ok. (The error comes up when checking convn in line 882 of ndstest).

01 Apr 2013 Wei

OK. I will try to find other ways to solve this issue. Thanks Matt.

31 Mar 2013 Matt J

Hi Wei. At minimum, you would probably need to upgrade to a 64-bit OS with more RAM. Your array is pretty big, though. It might be beyond the scope of what ndSparse can handle :(

31 Mar 2013 Wei

Hi Matt, when I ran command of[1e6,1e6,1e6]), I got message as following:
??? Error using ==> sparse
Sparse matrix sizes must be non-negative integers less than MAXSIZE as defined
by COMPUTER. Use HELP COMPUTER for more details.

Error in ==> ndSparse> at 2282

My computer is 32 bit windowsXP OS. Matlab version is 2010b. How do I solve this issue? Thanks.

05 Nov 2012 Matt J

@Racquel, I've fixed the indexing bug you reported (and one or two others). Thanks for letting me know.

17 Oct 2012 Raquel A

Thanks for the quick response, yes, I am aware I should be getting that, I just posted it as an example that somehow, the combination (58,58,...) gave an error but other ones that were close to 58 didn't.

Thank you for looking into it.

17 Oct 2012 Matt J

Hi Raquel,

When you do

X(57,57,1) = 1

you should be getting

X(:,:,1) =

(57,57) 1

This is what I get, so I hope what you posted was some sort of typo. As for the case of X(58,58,1) = 1, I will have to investigate that. Thanks for reporting it!

17 Oct 2012 Raquel A

Hi Matt, I have an issue when I run your code (thanks for submitting it though it's great)
Ok so for example I create a sparse matrix:[59,59,10])

somehow, there is an issue if I try to input in X(58,58,1). Every time I do this or any other number in the third dimension, it just fills up the full two first dimensions.

for example:

X(57,57,1) = 1

just gives me
X(:,:,1) =

(58,57) 1

BUT, if I try to do the same in

X(58,58,1) =1

I get

val(:,:,1) =

(1,1) 1
(2,1) 1
(3,1) 1
(4,1) 1
(5,1) 1
(6,1) 1
(7,1) 1
(8,1) 1
(9,1) 1
(10,1) 1
(11,1) 1
(12,1) 1
(13,1) 1
(14,1) 1
(15,1) 1
(16,1) 1
(17,1) 1
(18,1) 1
(19,1) 1
(20,1) 1
(21,1) 1
(22,1) 1


How can i fix this?

14 Sep 2012 Matt J

Hi Josh,

Please give me a code sample (a short one, ideally) that let's me reproduce this.

14 Sep 2012 Josh

Bug report: I have all zero ndSparse matrices that return a column vector for their size(). This causes all sorts of problems (cant' cat() them with otherwise same-dimensioned matrices, can't use full() on them...).

22 Aug 2012 Matt J

Hi Yoni,

Make sure that the 1st column of your PCoords data must range between 1 and 5

Similarly, the 2nd and 3rd columns of your PCoords data must range between 1 and 50.

The 4th and 5th column of your PCoords data must range between 1 and 100.

21 Aug 2012 Yoni Browning

Thanks so much for writing this!

I do just have one (hopefully) quick question: I would like to use your class to process a 5-dimensional dataset, lets call it 'P', such that size(P) = [5,50,50,100,100]. As per the recommendations for other commenters, I have been storing the coordinates of each non-zero point in P as the rows of one matrix (PCoords) and the values that should appear at each point in a sperate vector (PVals). Only at the end of my computation do I attempt to use these in a ndSparse object, which I have tried building both of these ways way:
P =,Pvals).
P =,Pvals,size(P),length(Pvals)).
Both of these causes the error:
Error using sub2ind (line 56)
Out of range subscript.

Error in ndSparse>ndCoords2IJ (line 3070)

Error in (line 2337)

Yes, I have checked, and PCoords and Pvals are the same length, dimensions, etc.

I am really sorry if I am missing something obvious, but I am having a hard time figuring out where I am going wrong from looking at your code, and I would really appreciate any direction you could give.

Thanks so much!

22 Jun 2012 Matt J

Hi Mehdi,

The problem is that your posts here are being received not just by me, but also by anyone who might be watchlisting this FEX file. This is SPAM to them. They expect to be receiving news from here only about ndSparse. I've moved your question to this Newsgroup thread:

22 Jun 2012 Mehdi

Hi Matt,

Thanks a lot for your quick reply and sorry about my lack of knowledge in this area. Since your area of expertise is image processing I thought you are the right person to ask. I need to create a 3D image dataset in which certain voxels (whose x,y,z values are my indices) will have a non-zero value.
I don't even mind if they are being assigned any value from 1 to 656535.
Because I can then segment them to two phases to get a 2-phase system.
Could you advise on what I can do to achieve this and get a 3D binary dataset made this way.

22 Jun 2012 Matt J

Hi Mehdi,

Since this no longer concerns ndSparse, you should probably post your issue in the Newsgroup. However, matrices and arrays are not functions. They can only be indexed by integers>=1. So, it is unclear what you mean when you say you want 1s assigned to non-integer indices.

22 Jun 2012 Mehdi

Hi Matt,

So many thanks for your constructive comments. It all worked fine and it indeed had the issues as you mentioned.

However, I am now facing a new issue. In a similar problem, I have to assign 1's to a 3D matrix with the x,y,z values of the coordinates of the points being single-precision floating numbers rather than intergers. Then span the same range of 0<=x<=1000 0<=y<=1000 0<=z<=1800 though.

When I tried the suggestions above it simply gives an error like:

??? Attempted to access b(39.372,36.681,44.517); index must be a positive
integer or logical.

Error in ==> matrix_3D at 24
b(a(i,c-2),a(i,c-1),a(i,c)) = 1;

Could you please advise any solution to this problem? I much appreciate your time.

19 Jun 2012 Matt J

Hi Mehdi,

I can tell you how to do this with ndSparse, but it sounds like your problem has simpler solutions.

First, you cannot use x,y,z coordinates that have value 0 since MATLAB indexing is 1-based. Doing so should have given you an error in your code. I'll assume henceforth that your mesh_data are really 1-based coordinates.

Second, since you say you have a powerful enough machine to hold a full double array of dimensions 1000x1000x1800 in MATLAB memory, it's not clear why simply changing the values from 0 to 1 pushes your memory limits over the edge. Changing the values of a nonsparse array doesn't even consume any extra memory. What's probably happenings is that your mesh_data are not really bounded to 1000x1000x1800 as you think, in which case MATLAB will try to grow the matrix to accomodate the new data. Growing the matrix would require extra memory and is also very speed inefficient. The solution, of course, would be to remove out-of-bounds coordinates from mesh_data or else pre-allocate a larger b to hold the coordinates you have.

Third, since your data consists only of 0s and 1s, you should probably store b as uint8, which would reduce the memory your code above is consuming by a factor of 8. You might want to see what happens if you make the following simple change to your code


In any case, to create the array you are talking about using ndSparse, you could do the following


If you have out-of-bounds mesh_data however this will not work

19 Jun 2012 Mehdi

Hi Matt, I am trying to create a 3D array of size 1000x1000x1800 with all the elements being zero and then assign 1 instead of 0 to some specific elements in the array and finally store it as a binary file.

Although MATLAB creates and saves the 3D array quickly, while trying to change zero's with one's , it runs out of memory.

((The 4,470,000x3 array named "mesh_data" is the coordinates (x,y,z) of the points that I need them to be assigned the value "1" in the 3D array.))

I should mention that all the x,y,z values of the coordinate matrix are non-negative integer numbers with: 0<=x<=1000 0<=y<=1000 0<=z<=1800

My ultimate aim is to export the 3D matrix (after all th esubstitutions), in a binary format (other than MATLAB's default binary format) so that I can use it with other programs.
Could you guide how I can use your program for this purpose?
Any help is much appreciated.

load mesh_data;
save('3d_matrix', 'b','-v7.3');
[r c]=size(a);
for i = 1:r,
b(a(i,c-2),a(i,c-1),a(i,c)) = 1;
save('mesh_matrix', 'b','-v7.3');

13 Jun 2012 Matt J

Josh - you mean you were doing an operation involving an ndSparse array and a full array of uint8s? And you were hoping to get a full array of uint8s as output? If so, I see what you mean. I'm considering redesigning the class so that it stores the non-zero values in tabular form, instead of as a MATLAB 2D sparse matrix. If I do, it will remove the need for mkCompat and you would get uint8 output or anything else you wanted. Until then, though, defaulting to doubles is the only way. A warning might not be a bad idea, though.

13 Jun 2012 Josh

In mkCompat, perhaps it would be worth doing some extra checks before defaulting to doubles, or at least a warning? Memory usage can go crazy at that spot. For various reasons, I had a huge array of uint8s. But it was only 1s and 0s. Converting that to a double made my memory usage spike to 8+GB in just a second or two. Let the user know that it was because they gave suboptimal inputs, not because your code is terribly inefficient.

14 May 2012 Wei

Thanks for the quick fix. Looking forward to your new solution.

11 May 2012 Matt J

@Wei: I've fixed the problem, but now I think I have a better solution. I'll post it over the next few days.

10 May 2012 Wei

found bug:
cat(2, ndSparse(rand(10,1,3)), ndSparse(rand(10,2,3)));

should be sth to do with the untrail in the permute ...

10 May 2012 Akram Bin Sediq

Thank you so much. This class should be added to MATLAB soon!

07 May 2012 Achim  
21 Apr 2012 Qiao Liyong

Thank you for your program.

12 Mar 2012 Matt J

Just as an FYI, the update a just submitted should make the kind of loops over slices G(:,:,i) go significantly faster. Here's a simple test I did,

>>[882,882,82]); Z=sprand(882,882,.001);

>> tic; for ii=1:82, G(:,:,ii)=Z; end; toc,
Elapsed time is 0.149033 seconds.

08 Mar 2012 J.D.

This works great. I love that so many functions are overloaded.

07 Mar 2012 Matt J

Hi Siavash,

If I've understood you, it sounds a lot like the issue in the comment posted above by Mircea (Aug. 27, 2011). If the application seems to hang, it probably hasn't stopped or crashed, but is probably taking a long time because of a sparse indexing bug in MATLAB (see my response Aug. 28, 2011).

The one thing that sounds strange in what you describe is that you say your loop runs fine up until the final iteration, kk_y. Is that correct? And if so, how do you know the earlier iterations run fine? I'll probably need to see your code to have the best picture of what's going on. You can send it through my author email link, if you like.

Beyond that, though, I want to repeat to you, what I told Mircea. Assigning into a sparse matrix (whether it be a normal MATLAB sparse matrix or an ndSparse object) is a bad practice and best kept to a minimum. Instead of loading data into the array slice-by-slice, you should generate a complete table of values in advance and then use it to build the whole 882x882x82 array in one single call to This is the approach I showed you in my comment posted yesterday.

07 Mar 2012 Siavash

Thanks for your prompt reply.
Actually i have a loop and in that loop I construct a 882by882 sparse matrix kk_y(82) times. and I want to store them in a 882*882*82 sparse matrix. but when it reaches to G(:,:,kk_y) = J_g_hat_kk;
it crashes! (no error, nothing, but only stop everything!).

06 Mar 2012 Matt J

@Siavash - your comment is a bit hard to interpret. You haven't said what kk_y is. Also, you say you're trying to make a "882 x 882 three dimensional sparse matrix", which seems self-contradictory since 882x882 would suggest 2D, not 3D.

However, if I had to guess, I would say that you're looking for this:


G =[ii,jj,kk],ss,[n_st n_st,kk_y]);

06 Mar 2012 Siavash

I tried to use your code to make a 882 by 882 three dimensional sparse matrix but it did not work and the computer was crashed. Simply this is what I do:
G =[n_st n_st]);
G(:,:,kk_y) = J_g_hat_kk;
J_g_hat_kk is a 882by882 sparse matrix. and n_st is 882. Could you please help me in this respect,
Thanks a lot

06 Mar 2012 Siavash


I tried to use your could to make a 882 by 882 three dimensional sparse matrix but it did not work and the computer was crashed. Simply this is what I do:
G =[n_st n_st]);
G(:,:,kk_y) = J_g_hat_kk;

J_g_hat_kk is a 882by882 sparse matrix. and n_st is 882. Could you please help me in this respect,

Thanks a lot

22 Feb 2012 Matt J

@Niklaus - sorry I didn't notice until now that this is happening for you in ndstest. Please post the entire error message so that I can know where in ndstest it occurs.

22 Feb 2012 Matt J

Hi Niklaus - the error you describe is unrelated to how new your MATLAB version is. The code is complaining that the reshape operation you're attempting is invalid because it doesn't preserve the number of elements in the array. Like trying to reshape a 2x5 array into a 3x3 array, for example.

22 Feb 2012 Niklaus

Hey Matt!
Sounds like a really neat code! Unfortunately I have an old version of MATLAB (R2009a). When I run ndstest, i get the following error message:

??? Error using ==> ndSparse>ndSparse.ndSparse at 126
The number of DATA elements is not consistent with the given ndShape.

Do you know why this happens? Is there any way for me to fix this? Thank you very much!


26 Nov 2011 Matt J

Incidentally, if you are now happy with the reimplementation I've made of the DISPLAY method, I would appreciate it if you would change your rating. One star out of five is, in any case, a harsh rating just because you don't like 1 out of the 92 methods offered by the class.

21 Nov 2011 Matt J

You talked me into it. I've now updated the distribution so that DISPLAY will print only those A(:,:,i,j,k,...) containing non-zero data.

For displaying purposes, at least, this should solve your problem. Mimicing the data you described, I'm now able to generate the following 37x37x37x37x37x37x37 data set on a 64-bit platform and get a reasonably instantaneous display:


However, I am vaguely concerned that you are pushing the limits of the class by having such a large array volume, even on a 64-bit platform. The class wasn't designed to handle arbitrarily large numel(A), just as ordinary MATLAB 2D sparse matrices were not.

I now clarify this a bit more in the description section of the FEX page.

18 Nov 2011 Matt J

@Jens: The reason I don't store my own tables is because then I couldn't piggy back on the fast sparse arithmetic routines of MATLAB's underlying 2D sparse class. I would have to implement my own.

In any case, the problem with displaying that you mention is probably not strictly related to data storage. It is related to the fact that the ndSparse DISPLAY method mimics the display method for full arrays and tries to print all "sheets" of the form A(:,:,i,j,k,...) to the screen. Since you have 37^5 sheets, naturally this is going to take some time to disaply to the screen.

Upon reflection, I admit that a sparser display would be more appropriate for an ndSparse class. For example, I could modify the DISPLAY method so that it only displays non-zero sheets.

On the other hand, as a designer, I can't really anticipate how every end-user would like their data displayed. Would a display of 2D sheets
A(:,:,i,j,k,...) offer meaningful information to someone working in 7-dimensional space, like yourself, even if I trimmed it down to the non-zero ones? And would every end-user prefer this?

Also, if you only want to display the non-zero data, why not use FIND? Or create your own wrapper for FIND which displays the data the way you want?

18 Nov 2011 Jens Munk Hansen

Hey Matt

Nice idea, but bad implementation. I am working with a parameter study and data is extremely sparse, dimensions are 37x37x37x37x37x37x37, but I have less than 400 non-zero elements. The display function takes more than 10 hours to execute.

Why not simple store tubles and use an intelligent search?

22 Sep 2011 Alexander

Illustration of my issue with sensitivity:

profile on
N=1e5; % data points
data=rand(N,1); % random data
ind=NaN(N,4); % random indices

% (100x100x100) x 200000 ind , data, dims);
% (100x200000x100) x 100 circshift(ind,[0 2]) , data, circshift(dims,[0 2]));

ans1=A1(1,:,:,:); % takes about 0.09 secs
ans2=A2(:,:,1,:); % takes about 115 secs !!!!

% results are equal w.r.t. dimension shift

profile viewer
% 99% of time is spent on a line data=builtin('subsref',,E);

22 Sep 2011 Matt J

This sounds different than Mircea's issue. If you can post a short example illustrating the problem, then I can possibly analyze it further

22 Sep 2011 Alexander

Ok, hope they'll fix this issue. For now I'll find a workaround.

Btw, for me the problem was not in assigning. I only create this matrix once (to store a very huge dataset), and then slice different data from it when I need. Sensitivity issue cames up on a line 478:


Anyway, thanks again, one of the best submissions I've seen on the exchange)

22 Sep 2011 Matt J

Hi Alexander, thanks for your feedback. I admit the shape sensitivity thing is a drag. However, the feature you describe would be very difficult to implement and require very pervasive changes throughout the code. I would hope that TMW will fix the shape sensitivity soon (it's already been reported) so that the issue will become moot.

I want to reiterate to you, though, what I said to Mircea. Assigning into a sparse matrix (or ndSparse object) is always best kept to a minimum, even without the shape sensitivity problem.

22 Sep 2011 Alexander

Hi, this class is just wonderful. However, if you consider making another release, there's one more feature it lacks.

I think you should give user ability to decide how to group matrices to store as 2D sparse. E.g. I need a 4-D matrix MxNxPxQ, but it turns out that due to shape sensitivity discussed above, the best best way for me to store it is (MxN)x(PxQ), otherwise performance decrease dramatically. Unfortunately, this currently can't be done in your class.
Thanks for such a great submission and looking forward to your reply

28 Aug 2011 Mircea

Thank you again, Matt J. It is clear I was going the wrong way working with sparse matrices (this is the first time I need it to use them). Thank you for your help.

28 Aug 2011 Matt J

@Mircea, the problem you're seeing is rooted in normal 2D MATLAB sparse matrices. It is related to the shape sensitivity issue being discussed in this thread:

There's not much I can do about it for now, but regardless, building sparse arrays the way you are doing is a bad idea, even for normal MATLAB sparse matrices. The efficient way is to create a complete table of non-zero entries first, and then construct the matrix using SPARSE or using

It is also a bad idea to use matrix assignment to alter the pattern of non-zeros in a sparse matrix, as illustrated by Tim Davis here:

27 Aug 2011 Mircea

Thank you, Matt J, this class is very usefull.

I have the following question: why does the following code

n = 500;
A =[2*n 2*n]);
A(1:2:2*n,1:2:2*n) = rand(n);

run in 0.04 seconds, while the following similar code

n = 500;
A =[2*n 2*n 2]);
A(1:2:2*n,1:2:2*n,1) = rand(n);

runs in 43 seconds (thats 1000 times slower!).

Is there a way to speed up the process of assigning values to truely N-dimensional sparse arrays (N>2)? Thank you.

14 Jun 2011 Matt J

@Ohad, I think that to fix it, you just need need to change the following line




I'll do another release once I've had time to examine this a little further.

14 Jun 2011 Ohad

I do not seem to be able to assign matrices into arrays. e.g.:

A(:,:,1) = sparse(ones(3))

any ideas?

27 Mar 2011 pink

how do I enter "ndSparse" into the program matlab
I use matlab2010a

28 Feb 2011 Matt J


My latest release includes the change of representation that we were talking about on Feb 10, 2011. That is, the data is now stored internally as a sparse matrix of dimensions (L_1*L_2*L_3...)xL_N. The memory savings should be quite noticeable.

Since the recoding was substantial, please backup your old version and give me feedback on any bugs that I may have introduced. Hopefully, ndstest was a pretty good filter, but you never know...

16 Feb 2011 Maigo

Thanks Matt :)

16 Feb 2011 Matt J

Maigo, thanks for your feedback. Your problem is indeed version-related. See my similar comment above to Farhad on 31 Jan 2011.

However, in the .zip file you should have found a version compatible with R2009b (or later). It contains isrow and iscolumn implementations and should have been fine for you. See also my comment on 04 Feb 2011.

16 Feb 2011 Maigo

Thank you very much. This class helped me a lot!

But I've found a bug with the "subsref" method, which calls two functions "isrow" and "iscolumn" that are not defined.

I am using Matlab 7.10.0 (R2010a), so I don't know if this is a version-specific problem. Maybe these two function are defined in 7.11.

Anyway, I think they're pretty easy to implement, and here's the code. Just add them to end of the class file:

function bool = isrow(A)
% isrow - returns true if A is a row vector
dims = size(A);
bool = (length(dims) == 2) && (dims(1) == 1);

function bool = iscolumn(A)
%iscolumn - returns true if A is a column vector
dims = size(A);
bool = (length(dims) == 2) && (dims(2) == 1);

10 Feb 2011 Bruno Luong

Check out this discussion about memory used by sparse matrix:

10 Feb 2011 Matt J

Mike, the explanation that I gave you was imprecise. See Tim Davis' explanation here,

He also gives links to expanded discussions on the subject.

10 Feb 2011 Mike

Matt: no problem. out of curiosity, how were you able to find out that matlab stores the # of elmnts for each column? on wiki it claims that matlab sparse uses COO format, which includes only the coordinates of each non-zero entry (so this is clearly a lie!). spinoffs of the Yale format seem very popular and do depend on the length of one of the dimensions and so seem a likely candidate, but they store something different from number of entries/column or row. when i go to sparse.mat i see only the help file, and the help file says nothing about what's going on behind the scenes, so just curious how you got to the bottom of it.

10 Feb 2011 Matt J

Mike: "But given that the size of matlab sparse matrices is proportional to the length of the 2nd dimension, I would vote strongly in favor of going to the (L_1*L_2*L_3...)xL_N sparse matrix representation..."

Yes, I've been thinking in that direction myself for a few weeks now. Unfortunately, it will require pervasive changes to the code, and will probably take me a few weeks at least :(

10 Feb 2011 Mike

Matt, thanks, that clarifies things. i still have a hard time thinking through how number of elements in each column helps on things like matrix multiplication, but perhaps i'm not being imaginative enough.

But given that the size of matlab sparse matrices is proportional to the length of the 2nd dimension, I would vote strongly in favor of going to the (L_1*L_2*L_3...)xL_N sparse matrix representation you mention unless there's a substantial slowdown in basic linear algebra. In the case of very large, very sparse, and very high dimensional matrices (of interest to me), the difference in storage space will be enormous.

10 Feb 2011 Matt J

"but you're saying the amount of memory of sparse matrices in both cases in linearly proportional to the total size of the matrix being represented??"

The main thing to understand is that every normal MxN sparse matrix internally stores a table of the number of non-zero elements in each column. In other words, it will always contain at least the same amount of data as in rand(1,N).

That's why the following two sparse vectors, even though they both contain all zeros, have very different memory consumptions.

>> A=sparse(zeros(1,1e6)); At=A.';
>> whos A At
Name Size Bytes Class Attributes

A 1x1000000 4000016 double sparse
At 1000000x1 20 double sparse

Pathological? Well, keep in mind that MATLAB sparse matrices were intended mainly for doing fast linear algebra. Apparently, these operations are faster if you know a priori the number of nonzeros in each column, instead of having to recompute them all the time on-the-fly.

10 Feb 2011 Mike

I see.. but then I'm confused why the memory consumed by normal sparse matrices doesn't depend entirely on the number of non-zero elements, as a sparse matrix is completely characterized by the coordinates and values of each non-zero element (plus,e.g., a pair of integers specifying the overall size in the 2D case). but you're saying the amount of memory of sparse matrices in both cases in linearly proportional to the total size of the matrix being represented?? if so, i think this is the far more pathological problem with matlab sparse matrices than them being limited to 2 dimensions!

10 Feb 2011 Matt J

Mike, just as a follow-up point, one thing that would reduce memory consumption is if I store the data internally as an (M*N)xP sparse matrix as opposed to a Mx(N*P) matrix.

Since MATLAB sparse matrices with fewer columns consume less memory, this would indeed help with memory consumption. I've been considering this for future releases, but I need to see how pervasive the effect would be.

10 Feb 2011 Matt J

Mike, the amount of memory consumed by an MxNxP ndSparse array A, will essenitally be the same as the ordinary 2D MATLAB sparse matrix of dimensions
Mx(N*P) generated by doing

You can see this better in a larger version of your example,


>>whos A As
Name Size Bytes Class Attributes

A 20x40x60 9696 ndSparse
As 20x2400 9616 double sparse

In fact, the way it works is that the object A is storing its data internally as the 2D matrix As. There is a difference of about 80 bytes due to other minor ndSparse class data.

So the answer to your question is, there is no memory inefficiency in ndSparse that isn't in normal MATLAB matrices. Since mine is built on top of theirs, there's not much I can do about it.

One thing you should be aware of, though, is that the memory consumption of sparse matrices, both mine and MATLAB's, is shape-sensitive. Permuting the dimensions can significantly reduce memory consumption, e.g.,

>> Ap=permute(A,[3 2 1]); whos Ap

Name Size Bytes Class Attributes

Ap 60x40x20 3296 ndSparse

10 Feb 2011 Mike

if i do "a = ndSparse([])" and then "a(1,2,3)=4", i get back:
"a(:,:,1) =
All zero sparse: 1-by-2
a(:,:,2) =
All zero sparse: 1-by-2
a(:,:,3) =
(1,2) 5"
each of these "all zero sparse" matrices take up memory and don't seem worth keeping track of. is it possible to eliminate this behavior?

04 Feb 2011 Matt J

@Farhad, I've now included a version of ndSparse that is compatible with R2009b. See the UpdateNotes.txt file for details.

31 Jan 2011 Matt J

@Farhad - You don't have a recent enough version of MATLAB, and so are missing some of its newer functions. You should upgrade to R2010b. Alternatively, you could write your own iscolumn.m

function bool=iscolumn(v)

bool=isequal(size(v), size(v:));

I can't be sure that that will be the only missing function, but I'm pretty optimistic...

31 Jan 2011 Farhad M

Hi, thanks! it must be very much useful for my purpose. However, I couldn't run ndstest since it gives the following error:

??? Undefined function or method 'iscolumn' for input arguments of type 'ndSparse'.

Could you help me with that?

11 Jan 2011 Matt J

@vikranth, I'm not sure I understand your question. What details are you looking for beyond the description above, the GettingStarted file in the .zip, or in the help doc?

11 Jan 2011 vikranth akkidas

can u give some details about the topic

10 Jan 2011 Michael Völker  
04 Jan 2011

*Some improvements to CAT and SUM, including minor bug fixes.
*Added methods MIN, MAX, ANY, ALL for the class.
*Improved error checking.

30 Jan 2011

See UpdateNotes in the .zip for full details. Highlights:

1. Lots of new methods!! For example BSXFUN, REPMAT, CIRCSHIFT,CONVN.

2. Improved the efficiency of PERMUTE

3. Some small backward-compatibility issues.

04 Feb 2011

See UpdateNotes in the .zip for full details. Highlights:

1. Lots of new methods!! BSXFUN, REPMAT, CIRCSHIFT,CONVN,...
2. Improved the efficiency of PERMUTE
3. Some small backward-compatibility issues.
4. Added an R2009b compatible version.

07 Feb 2011

Fixed a bug in the constructor, which could cause ndSparse object to have incorrect dimensions.

27 Feb 2011

Pervasive changes, some new methods, and some bug fixes. See UpdateNotes in .zip for full details.

Among other things, ndSparse objects should typically consume a lot less memory now.

28 Mar 2011

Fixed a bug in SUBSASGN operations that expand the size of the matrix,
e.g., A(end+1,end+1)=5;
Thanks to Zvi Devir for reporting.

05 Jul 2011

*Some bug fixes, including Ohad's issue of assigning a matrix into an array.

*Improved memory efficiency of various operations, but with trade-offs. See UpdateNotes

07 Jul 2011

*Forgot to add installation instructions in my submission earlier this week
*Also, replaced ISDOUBLE in ndsparse.accumarray with something more universal.

21 Nov 2011

Improved the DISPLAY method, so that now only A(:,:,i,j,k,...) with non-zero data will be displayed.

21 Jan 2012

Fixed the output of and CONVN, which sometimes had very large nzmax.

11 Mar 2012

Speed up indexing operations of the form A(:,:,...,:,i)

11 May 2012

Fixed bug involving ndSparse arrays with singleton dimensions. It affected MIN, MAX, SUBSASGN, and CAT.

22 May 2012

Improvements to MIN/MAX when dealing with singleton dimensions.

Improved memory efficiency in CAT and null assignment.

05 Nov 2012

Indexing bug fixes.

13 Mar 2013

Indexing bug fix. Affected scalar, linear indexing (see update notes).

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